Design of experiment-based tolerance synthesis for a lock-or-release mechanism of the Chinese Space Station Microgravity Platform

Design of experiment-based tolerance synthesis for a lock-or-release mechanism of the Chinese Space Station Microgravity Platform

Design of experiment-based tolerance synthesis for a lock-or-release mechanism of the Chinese Space Station Microgravity PlatformDesign of experiment-based tolerance synthesis for a lock-or-release mechanism of the Chinese...Jian Ding et al.

This paper deals with the tolerance synthesis with the application
for a typical Lock-or-Release (L/R) mechanism, used for Chinese Space Station
Microgravity Platform (SSMP). The L/R mechanism is utilized to lock the SSMP
maintaining space position during the launching stage, and to release the
SSMP automatically during on-orbit stage. Manufacturing accuracy of L/R
mechanism presents direct influence on its kinematic and dynamic behaviors.
Tolerance synthesis can provide a reasonable assignment of tolerance,
satisfying the critical assembly criteria while lowering manufacturing
complexity. In this paper, based on the number-theory method (NTM), a
Halton-set based Monte Carlo (MC) simulation is introduced in the accuracy
model of the L/R mechanism, aiming at improving analytical precision and
efficiency for tolerance synthesis. A design of experiment (DOE) based
tolerance synthesis approach is proposed. With initial tolerance determined
by capacity, sensitivities of different tolerance factors are generated
through the first DOE stage, and then applied to determine feasible tolerance
levels. The final tolerance assignments, like points scatted in
high-dimensioned space with inherent uniformity, are then produced through
uniform DOE in the second stage. Result shows that the majority of feasible
tolerance assignments generated have more relaxed tolerance, which can
facilitate the manufacturing process.

As one of scientific exploration oriented carriers, the Chinese Space Station
Microgravity Platform (SSMP) provides a higher level of controllable
environment for samples and instruments and facilitates delicate
manipulations in diverse new technology experiments, such as material
sciences, microgravity fluid physics and biotechnology (Xie et al., 2016;
Wang et al., 2014; Liu et al., 2006). The L/R mechanism is configured in
parallel, developed and equipped supportively for the SSMP, and its
hierarchical relation is illustrated in Fig. 1. The conceptual model of the
Chinese Space Station, consists of a core chamber module and four
experimental chamber modules. In each experimental module, each scientific
research experiment rack is fixed in parallel with the others (Zhou, 2013;
Liu et al., 2014). Two auto L/R
mechanisms are assembled into both sides of each scientific research
experiment rack to lock the SSMP. During the shuttle launching stage, the
SSMP is securely locked by eight lead screws of the L/R mechanisms on both
sides. While arriving at the scheduled orbit of the space station, the lead
screws on each side are driven inversely to release the SSMP for microgravity
experiments.

Figure 1Hierarchical relation (a) conceptual model of the Chinese
Space Station; (b) scientific research experimental rack and the
SSMP with two L/R mechanisms in a locked status; (c) physical
prototype and (d) 3-D model of the L/R mechanism on a side of the
SSMP.

Manufacturing accuracy is critical to the kinematic and dynamic performance
of the L/R mechanism (Ding et al., 2018). Nowadays, lower consumption and
higher performance have been always pursued for all manufacturing companies,
particularly considering the current context of increasing global
competition. Tolerance assignments for mechanical parts and assembly of a
product play an important role in the accuracy performance of the product,
since it is closely connected with components' tolerance (Merlet, 2006; Hao
and Kong, 2016; Hao and Merlet, 2005; Huang et al., 2016; Huang and Kong, 2010; Li et al., 2016). As one of
the crucial tasks in a product life cycle, tolerance synthesis for mechanical
parts and assembly of a product, which is generally regarded as tolerance
design, is a typical iterative procedure. It ranges from tolerance
initialization to final reasonable tolerance determination, in which
constraints such as manufacturing capability, performance quality and
production cost are considered (Moroni and Polini, 2003; Pasupathy et al.,
2003). There are two problems to be addressed: (a) how the mechanism
tolerance is assigned, so that manufacturing complexity is reduced without
much loss in quality (Singh et al., 2009a; Chen and Ji, 2005; Chlebus and Wojciechowska, 2016; Jeang,
2001; Lyu et al., 2006; Rout and Mittal, 2006, 2007, 2008; Li et al., 2015);
and (b) how the synthesis approach is devised so that process data can be
promoted in precision and efficiency (Singh et al., 2009b; Huang, 2013).

1.1 Mechanism tolerance synthesis review

Parallel-type mechanisms (Merlet, 2006; Hao and Kong, 2016; Hao and Merlet,
2005) are analogous to the L/R mechanism in structure. Huang et
al. (2016) proposed a comprehensive
methodology for a 4-degrees-of-freedom (DOF) high-speed pick-place parallel
robot with an articulated travelling plate, where the tolerance model for the
tilt angular error was established statistically. A strategy was to minimize
the total cost while satisfying the permitted angular error and manufacturing
feasibility constraints, and a reasonable angular accuracy within cylindrical
task workspace is finally generated. Wang and McCarthy (2018) designed a
four-bar function generator to act as a flapping wing mechanism. The
tolerance zones were specified around the accuracy points, and twenty-nine
designs were found to achieve the desired coordination of wing's swing and
pitch. Yin et al. (2018) investigated tolerance design for spatial mechanisms
with the use of both the extremum and the probability methods. In comparison
with several surrogate models, the tolerance was optimized and thus
manufacturing cost is significantly reduced.

DOE based tolerance design has drawn close attention in recent years for
engineering applications. Employing the concept of Taguchi S/N ratio, Rout
and Mittal (2006, 2007, 2008) utilized the inner and outer orthogonal array
to identify significant parameters of a 2-DOF planar manipulator with two
revolute joints for optimum tolerance. Optimum tolerance of the manipulator
with payload was finally allocated, and was validated by Monte Carlo (MC)
simulation. Huang (2015) developed a Taguchi based optimum tolerance
design for a function generation mechanism with joint clearance, where the
total cost of assembly was minimized while satisfying the accuracy
requirements. Insight on parameter variances of this mechanism was provided
with the sensitive maps, and the efficiency and practicality were
demonstrated with the proposed method. Li et al. (2015) investigated the
tolerance design problem of a 6-DOF space docking mechanism using uniform
design, and concerned the efficiency and precision of MC simulation in DOE.
The tolerance of the component in the mechanism was properly broadened
without any loss of output accuracy, and the manufacturing cost was reduced
to a certain extent. Huang (2004, 2013), Huang and Kong (2010) and Zhou (2001) made a comprehensive introduction to the
advantages of number-theoretic method (NTM), and exhibited the significant
precision and surprising efficiency of NTM in tolerance design and process
simulation, which made the comprehensive theory of NTM start to serve the
manufacturing in engineering.

1.2 DOE based tolerance synthesis methodology

In a process for mechanism manufacturing, the capacity is constantly
influenced by multiple stochastic factors, and presents fluctuant. The
tolerance assigned may not be possibly dynamically suitable in practice.
Therefore, tolerance assignment should be with the characteristics of
redundancy and flexibility. Tolerance synthesis approach should have high
analytical precision and efficiency, and generate tolerance assignments with
completeness and representativeness, that could be actively resistant to
potential failure, due to instability of manufacturing capability.

NTM offers a possible solution for this problem. The essence of NTM is to
determine a point-set in a s-dimensioned super-cube, where the points in
the set are uniformly distributed (Fang and Wang, 1994). Till now
mathematicians have proposed several types of set such as Hua–Wang set (Hua
and Wang, 1972), Halton set (Halton, 1960; Chi et al., 2005), Niederreiter set (Niederreiter,
1992) and Sobol set (Sobol, 1967; Bratley and Fox, 1988). They are all termed as low discrepancy
sequence (LDS). The initial target of LDS aims at improving MC simulation
performances, which are prevailing over the traditional pseudo-random set,
then the LDS is introduced in the field of DOE, and as the basis, uniform DOE
is established (Fang and Wang, 1994) with GLP-set (Hua and Wang, 1972).
Uniform DOE pays more attention on distributional uniformity other than
symmetrical comparability of experiment points. It could disclose system
information with the fewest representative points. On the contrary, the
experiment points construct an optimum and uniform coverage of experiment
space.

With this advantage, the uniform DOE array, composed of different levels of
tolerance in a mechanism, could construct plenty of candidate tolerance
assignments. They are optimally and uniformly distributed within the
experiment space. As manufacturing capability fluctuates, these assignments
can be valuable and robustly adapt to variation of manufacturing capability.

In this paper, the accuracy model for the L/R mechanism is established
firstly. For a precise and efficient simulation process, the Halton-set based
MC simulation, is introduced in tolerance analysis of the L/R mechanism.
Then, tolerance synthesis is proposed consisting of two stages. The first
stage of DOE, provides tolerance sensitivity, and the second one, applies
uniform DOE to generate tolerance assignments with representativeness and
uniformity, against capacity instability and manufacturing complexity. The
flowchart of the tolerance synthesis applying both NTM and uniform DOE is
illustrated in Fig. 2.

Building on the above advances, this paper focuses on tolerance analysis and
synthesis towards application in the L/R mechanism. The reminder of this
article is organized as follows: Sect. 2 briefly introduces the structure and
principle of the L/R mechanism; Sect. 3 completes accuracy modeling for the
L/R mechanism, and conducts tolerance analysis with Halton-set based MC
simulation; Sect. 4 details the tolerance synthesis procedures for the L/R
mechanism and illustrates the practicability for a case study; Conclusions
are drawn in Sect. 5.

Figure 2Flowchart of tolerance synthesis with both NTM and uniform DOE.

The architecture of a L/R mechanism is illustrated in Fig. 3. The stepper
motor on the backside, drives the bevel gear Z1, then revolve gears Z2 and Z3
that are connected together. Driven by a central gear Z4, four branches of
gears rotate simultaneously, making four lead screws move forward as
synchronous as possible for the locking gears Z5, Z6, Z7, and Z8. The SSMP is
securely locked by contacting the ends of the four lead screws, with the
slots on both sides of the SSMP. While arriving at scheduled orbit, four lead
screws on each side are then driven inversely to release the SSMP.

The L/R mechanism with imperfect manufacturing quality results in two
problems: firstly, inaccurate location and transmission of any lead screw
would cause the contact backlash between the SSMP and each lead screw. It
tends to raise impact damage structurally by harmonic response of the SSMP's
vibration during the launch stage; Secondly, the unbalanced locking supports
on the SSMP inversely make lead screws unable to form symmetric or regular
deformation, which also poses a potential threat to the shuttle structurally.
However, higher manufacturing accuracy for the L/R mechanism would inevitably
increase manufacturing complexity; it may be unreachable to manufacturing
capability. Therefore, an optimum compromise between tolerance and capacity
for the mechanism is essential to vouch the system's high reliability and
security.

Geometrical errors of the L/R mechanism come from the uncertainty of
connected components' position and orientation during the manufacturing
stages, and complicate the influence of the output errors, which are
represented by the end errors of the lead screws. In this section, an
accuracy model of the L/R mechanism using vector differential algorithms is
established, which provides linear relations between geometrical errors and
output errors. On improving accuracy simulation precision and efficiency, the
Halton-set based MC simulation is introduced and compared to the traditional
pseudo-random set based MC simulation.

3.1 Mechanism accuracy modeling

Both the L/R mechanisms are symmetrically configured on both sides of the
SSMP as shown in Fig. 1. Each side can provide enough geometric information
for accuracy analysis. Four lead screws are fixed on the backplane of the L/R
mechanism, and they are guided by a gear train to synchronously travel
forward until their ends contact slots of the SSMP on a side.

The imperfectness of assembly generates locational errors δxp,
δyp, δzp, δα, δβ and δγof the backplane, they are with respect to the nominal position of
the scientific research experiment rack; similarly, each lead screw on the
backplane has locational errors δai,x, δai,y, δai,z, and angular errors δui,x and δui,y about x
and y axes. The axial error δl of each lead screw is theoretically
contributed by transmission errors from the engaged gears and assembly error.
However, repetitive experiments had indicated the axial error of lead screw
assembled weighted majority among the translation errors, therefore, the
transmission errors are not included in the mechanism accuracy model. The
output errors δbi,x, δbi,y and δbi,z refer
to inaccuracy between the end of each lead screw and nominal contact center
on a side of the SSMP.

Frames {OB} and {OA′} attaching at the nominal centers on a
surface of the SSMP and mechanism's backplane, are outlined in Fig. 4
respectively. Frame {OA′} has a positional error
vector δp (δxp,
δyp, δzp)T and an angular error vector δΩ (δα, δβ, δγ)T with respect
to the base frame {OB} . Four lead screws in both coordinate frames
form four closed kinematic loops. Since all the lead screws are
centrosymmetric about the z′ axis of coordinate frame {OA′} , either
closed-loop kinematic chain is established independently and
representatively.

Figure 4Assembled L/R mechanism with the SSMP and kinematic diagram for this
simplified L/R mechanism.

In Fig. 4, the closed-loop kinematic chain OB-OA′-Ai′-Bi′ can
be expressed with a vector equation as

(1)bi=p+Rai+liui(i=1,2,3or4)

where R refers to the rotational matrix of frame {OA′} with
nominal orientation angles α, β and γ, about x, y
and z axes, and can be written as

R=cosγ-sinγ0sinγcosγ0001cosβ0sinβ010-sinβ0cosβ1000cosα-sinα0sinαcosα

Differentiating both sides of Eq. (1) yields

(2)δbi=δp+δRai+li⋅ui+Rδai+δli⋅ui+li⋅δui

where δbi (δbi,x, δbi,y, δbi,z) denotes the end error vector of theith lead screw in frame
{OB} ; δp(δxp, δyp, δzp)T refers to the positional error vector of the backplane with
respect to base frame {OB}; δR denotes perturbation of
the rotational matrix R of frame {OA′} with regard to the
base frame {OB}; ui (ux, uy, uz)T refers
to the unit vector of the ith lead screw; δui represents a
deviation of ui, and can be expressed as

(3)δui=Δuiui=0-δuzδuyδuz0-δux-δuyδux0uxuyuz

where Δui denotes the antisymmetric tensor of δui.

δR in Eq. (2), has to deal with the perturbation vector
δΩ (δΩx, δΩy, δΩz)T of the nominal angles α, β, and γ, with
respect to base frame {OB} . Therefore, δΩ can be
detailed as

Since the nominal orientation angles α, β, and γ of
the frame {OA′} with respect to base frame {OB} are all zeros
in the geometric configuration of the SSMP and the L/R mechanism, δR can be simplified as

(4)δR=ΔRR=0-δγδβδγ0-δα-δβδα0R

where ΔR is an antisymmetric tensor of δΩ; let ci=R(ai+liui),
Eq. (2) can be rewritten in a compact form as

(5)δbi=EΔciTδpδΩ+R⋅δai+δli⋅ui+li⋅Δui⋅ui

where E is a 3×3 unit matrix; Δci is an
antisymmetric tensor of vector ci.

The nominal parameters in Eq. (5) can be determined as follows: the nominal
orientation angles α, β, and γ of the backplane are
set zeros, therefore the orientation matrix R becomes a 3×3 unit matrix; The unit vector ui with nominal value (0,0,-1)T for each lead screw synchronizes with frame {OA′} while micro
rotating of frame {OA′} occurs. The x and y components of vector
δui can be approximated with micro rotational angles,
Δθ1 and Δθ2, for a lead screw about its own
x and y axes, respectively, and z component of ui is zero.
Thus, Eq. (5) can be unfolded as follows

3.2 Halton-set based MC simulation

The technique of MC simulation has been received continual recognition in
engineering practice. As one of the diagnostic process in tolerance design,
MC simulation provides the statistics of output errors that assist the
designer to make reasonable tolerance design of a product. However, there are
two main problems for MC techniques: (a) time consumption for a large scale
problem is unaffordable; and (b) solution precision for a median problem is
unacceptable. Therefore, improving the precision and efficiency for a MC
simulation is essential.

In recent decades, the NTM gradually became popular in computational
mathematics area, and prevailed over traditional MC techniques in precision
and efficiency. Instead of pseudo-random set in statistical simulation, the
NTM applies LDS, whose points scatter evenly in a unit cubic, and whose
regularity can be evaluated mathematically. The NTM based simulation could
provide a higher convergence rate induced by its computation complexity of
O(logNs/N) than that induced by the one of O(N-1/2) provided by
pseudo-random based MC simulation, where s represents the dimension of the
investigated problem.

For Halton sequence (Halton, 1960), each point k can be represented by a
m-ary expansion:

(7)k=b0+b1m+b2m2+…+brmr

where m is a prime number less than integer bi,0<bi<m-1,
i=0,1,2…r. The component of each point in Halton-set can be
represented by a radical inverse function ym(⋅) defined as

(8)ym(k)=b0m-1+b1m-2+…+brm-r-1

where ym(k)∈(0,1), and the each point, Xk, in Halton-set with
s-dimension, can be expressed as

(9)Xk=ym1(k),ym2(k),…,yms(k)

where a set of pairwise coprime m1, m2,…,ms are selected
as dimensional bases. Generally, the first s prime numbers are favorable in
simulation.

3.3 Effectiveness comparison

A comparison to output errors of the L/R mechanism for traditional
(pseudo-random based) and Halton-set based MC simulation, are performed. All
the parameters for the L/R mechanism are listed in Table 1. All the geometric
errors in Eq. (6) are presumed to conform to uniform distribution within
their tolerance zone, and listed in Table 2. The simulation comparison
employs the sample size of 0.5×103, 2.0×103, 4.0×103, 6.0×103, 8.0×103, 1.0×104, and
1.2×105. Standard deviations of output errors are represented by
the x, y and z error of the end of each lead screw and listed in
Table 3. Variations of standard deviation convergences are shown in Fig. 5.

The relative error σR is selected to evaluate the precision between
two approaches and defined as

(10)σR=σS-σTσT×100%

where σS represents standard deviation simulated with either
approach, and σT is theoretical standard deviation with
analytic error model in Eq. (6).

Figure 5 illustrates that the standard deviation σΔbi,x,
σΔbi,y, or σΔbi,z of the L/R
mechanism' output errors with Halton-set based MC simulation, has a higher
convergence rate than those with pseudo-random based one. The theoretical
values of the statistics σΔbi,x, σΔbi,y, or σΔbi,z, are processed with the L/R mechanism
error model in Eq. (6), and based on the nominal parameters in Table 1 and
parameters' errors in Table 2. To reach the same simulation precision, for
instance, we simply define a relative error of σR≤0.1 %. For
σΔbi,x, it is achieved with a sample size of 500 by
Halton-set. It is smaller than that of 1.2×105 by pseudo-random
based MC simulation; for σΔbi,y, it is achieved with a
sample size of 2×103 by Halton-set, and it is smaller than that of
4×103 by pseudo-random based MC simulation; for σΔbi,z, it is achieved with a sample size of 500 by Halton-set, and it is
also smaller than that of 2×103 by pseudo-random based MC
simulation. Additionally, the convergences for σΔbi,y and
σΔbi,z with pseudo-random based MC simulation fluctuate
significantly. In the following section, the tolerance synthesis process is
therefore performed with Halton-set based MC simulation for responses of DOE.

We have found that as the dimension of a problem increases, the priorities in
analytical precision and efficiency based on NTM are more significant over
traditional Monte Carlo simulation. This simulation belongs to a 12-dimension
problem, and Li et al had applied Sobol-set based MC in tackling accuracy
analysis problem of a six-dof docking mechanism with 42 dimensions (Li et
al., 2015). as the dimension increases, the pseudo-random cannot guaranty the
investigated space where the high-dimension points uniformly are scatted.
Additionally, there are some differences in the construction methods of
different point sets. With the deepening of the number theory research, point
set construction methods to improve the accuracy and efficiency of analysis
will continue to emerge.

Taguchi (Rout and Mittal, 2006, 2007, 2008) suggests that the quality of a
product is not ensured in the checking stage, but determined in the design
stage. A reasonable tolerance assignment could not only facilitate production
process but also bring about better performance of a product.

Generally, traditional tolerance synthesis for a mechanism is considered as a
typical optimization technique. It aims at achieving a compromise between
tolerance and cost, under constraints of various quality criterions. The
optimized tolerance is then directly applied in the mechanism manufacturing,
without much modification. This is currently a general process for mechanism
development in a laboratory.

However, in contemporary manufacturing enterprise, there are plenty of
stochastic factors disturbing process capacity dynamically; traditional
synthesis approach is not perfectly suitable to modern production. It is
necessary to propose an enterprise oriented mechanism's tolerance synthesis
approach, which generates tolerance assignments with enough flexibility to
adapt to stochastic variations of capacity.

4.1 DOE based synthesis methodology

In view of DOE, different tolerance as factors in a mechanism can be
considered as independent dimensions. They are spanning a multi-dimensioned
space, which can be regarded as “tolerance space”. In this space, the
boundaries can be estimated by manufacturing capability, potentially feasible
tolerance assignments are enveloped.

In tolerance space, different tolerance has their corresponding sensitivities
to mechanism output performance. Tolerance sensitivity can be provided
through range analysis. Then we can use their sensitivities to update levels
of different tolerance, as level variation of any tolerance contributes
equivalent influence on the mechanism output performance.

With the newly updated levels, how they can be merged into expected tolerance
assignments, is an interesting and practical problem. An intuitive idea is to
enumerate all the levels of all the factors, however, as the number of
tolerance factor or levels increases, the sorting process may possibly result
in a combinational explosion.

A compromise approach is to select representative combinations of levels in
tolerance space. The representative combinations, as tolerance assignments,
can be determined by a uniform DOE array, with good uniformity and neat
comparability. They can optimally cover the tolerance space and be tolerant
to capacity variation well.

The uniform DOE, short for uniform design, is proposed in the 1980s by Fang
and Wang (1994), and is distinct from typical
orthogonal design and Latin square design. The uniform experiment points are
uniformly scattered within the whole experiment space, and fewest
representative experiment points can disclose most information of a system,
other than the rest types of DOE.

Uniform DOE (Fang and Lin, 2007) is particularly proficient in tackling high dimensional problem
with factors and levels of large numbers. Uniformity of a uniform DOE can be
measured through L2-discrepency as follows:

Given a s-dimensioned super-cube Cs=[0,1]s, a set of experiment
points P={x1,…,xn} is constructed and distributed within the
entire experiment space as uniformly as possible. L2-discrepency, short
for CD2(⋅), is used to evaluate the uniformity of the point set P and
can be written as follows

where xk=(xk,1,…,xk,s) is the kth experiment point. In
terms of the point set generation rule, the experiment space is consistently
filled with uniform points of different quantity, therefore, the set of
tolerance assignments, as experiment points, can be ultimately formulated.
With a proper uniform design array, a tolerance assignment set can also be
available.

4.2 DOE based synthesis procedure

In the first stage of DOE, a proper DOE array is firstly selected in terms of
the tolerance factor. Tolerance levels are initially evenly divided with
geometric tolerance determined by manufacturing capacity, since we do not
have any knowledge of their sensitivity. Then Halton-set based MC simulation
is employed for DOE response, please refer to Sect. 3.2. Through range and
variation analysis techniques, the sensitivities of geometric tolerances are
subsequently obtained. Figure 6 detailed the scheme of the first stage of DOE
for factor sensitivities, which mainly includes preparations for 1-DOE,
tolerance response simulation and tolerance sensitivity analysis.

The second stage of DOE aims at generating uniformly distributed tolerance
assignments. Sensitive factors are set with the highest level, the rest
insensitive ones are re-divided into levels, whose intervals are inversely
proportional to their sensitivities, and therefore levels of tolerance
factors can produce an equivalent impact on response. In this stage, DOE
responses are also generated by Halton-set based MC simulation, please refer
to Sect. 3.2.

4.3 Definition and Strategy

DOE based tolerance synthesis for a mechanism has to: define response
function, arrange tolerance factors and levels, and devise other detail
strategies related to tolerance assignment. They are addressed as follows.

Figure 6Scheme for the first stage of DOE for the L/R mechanism's tolerance
synthesis.

4.3.1 Response function definition

The tolerance stack-up is used to reflect the success rate of the assembled
mechanism, and evaluate manufacturing quality whether this tolerance
assignment is acceptable. It involves two aspects: radical error Δri of each lead screw and non-synchronous error Δz of all 4 lead
screws of the mechanism, they can be respectively defined as

where Δxp,i and Δyp,i denote the end errors of a lead
screw along x and y axis respectively, and can be obtained from Eq. (6).
Δr0 represents the allowed threshold of a radical error in a side
surface of SSMP in Fig. 4. Δz0 is the permitted threshold of
non-synchronous error.

It is noted that there is a statistical process. We define an event
“Mi” as an occurrence of the radical error Δri of the ith
screw's end, within the threshold of Δr0, which can be expressed
as Δri<Δr0. Then another event “N” is defined as an
occurrence of non-synchronous error along z axis among the ends of screws,
within a threshold of Δz0, which can be expressed as “Δz<Δz0”. The tolerance stack-up is the probability of all the
events “Mi” (i=1, 2, 3, or 4) and “N”, occurring at the same time.
Thus, the tolerance stack-up can be expressed as a probability of multi-event
production, since they are independent with each other:

We can take 99.73 % as a threshold of the tolerance stack-up, in terms of
“6σ” principle in production quality control theory. It means if a
combination of tolerance levels in the uniform DOE yields a response
P(M1M2M3M4N)>99.73 %. This combination can be
accepted as one of potentially feasible candidate tolerance assignment. A
detailed schematic for simulation process is outlined in Fig. 7.

4.3.2 Arrangements for factors and levels

In a tolerance synthesis, geometric tolerance of a mechanism as factors are
considered before a proper DOE array is determined. The first stage of DOE is
designed to obtain a general knowledge of sensitivity of different tolerance
factor. Therefore, over-condensed levels are not necessary for DOE analytical
efficiency.

4.3.3 Strategy for level re-division

For the second DOE stage, interval between adjacent levels of a tolerance
factor follows inverse proportion to its own sensitivity, as to place an
equivalent impact on mechanism output performance. The level re-division
follows three steps:

a.

Identifying the most insensitive factor, and conserving its sensitivity as
R; the interval Δi between adjacent levels of the rest factors is
expressed as:

(15)Δi=RiR×Thi-Tlin

where Ri denotes the sensitivity of the ith tolerance factor;
Thi and Tli denote the upper and lower bounds of the ith
tolerance factor, respectively, n denotes the expected number of levels.

b.

Determining new levels of all factors as:

(16)Li,j=Tli+Δi×(j-1)0<j<Thi-TliΔi

where Li,j refers to the jth level of the ith factor, and j is an
integral number.

c.

Simulating with Halton-set based MC approach with these newly updated
levels, and yielding tolerance stack-ups as responses.

Figure 7Scheme of Halton-set based simulation for the tolerance stack-up and
for each column in the uniform DOE array.

4.4 Illustrative example

The L/R mechanism discussed is shown in Fig. 4, the whole tolerance synthesis
is presented as follows:

4.4.1 Parameters initialization

Quality index requirement

The radical error Δr0 defined in Eq. (12) is set with 1.6 mm,
which means radical deviations for all the screw ends are restricted within
1.6 mm. The non-synchronous error Δz0 defined in Eq. (13) is
configured with 1.6 mm. To ensure the manufacturing quality, the tolerance
stack-up should be no less than 99.73 % in terms of “6σ”
principle.

Traditional tolerance synthesis approach comes from experience in practice,
does not place sufficient concern on the sensitivity of different tolerance.
That makes the tolerance assignment become less flexible, and limits the
assembly application. In terms of experience, the tolerance assignment is
obtained and listed in Table 4, and whose stack-up is processed with the
scheme described in Fig. 7, and finally reaches up to 99.99 %
(> 99.73 % of “6σ”). This tolerance assignment is acceptable
for a successful assembly of the L/R mechanism, however, as manufacturing
capacity varies, its feasibility is still doubtful.

Tolerance factors' initialization

All the geometric tolerance factors are initialized as follows: the permitted
limitations of dimensional tolerances involving Δxp, Δyp, Δzp, Δai,x, Δai,y, Δai,z,
Δl are no less than ±0.1 mm; and permitted those of angular
tolerances involving Δα, Δβ, Δγ,
Δθ1, Δθ2 are no less than ±0.1∘.
Since the technological process tolerance Δai,x for a lead screw,
resembles that of Δai,y, they are regarded as the same factor;
similarly, the same analysis is for Δθ1 and Δθ2 which can also be regarded as the same factor. Therefore, 10
tolerance factors are finally determined: Δxp, Δyp,
Δzp, Δα, Δβ, Δγ, Δai,x (or Δai,y), Δai,z, Δθ1 (or
Δθ2) and Δl.

Table 7Variance analysis for the sensitivities of the tolerance factors.

4.4.2 The first stage of DOE

With DOE based tolerance synthesis discussed above, the orthogonal array
L27(310) is chosen for the first stage of DOE. It accommodates 10
factors, three levels for per factor and 27 combinations, can be considered
as an appropriate arrangement for compromise between efficient and precise
simulation, and is listed in Table 5.

With the first DOE in Table 5, the responses are obtained through a sample
size of 105 Halton-set based MC simulation and tabulated in Table 6.

For precise tolerance sensitivity in the first stage, a convergence
comparison with pseudo-random and Halton-set based MC simulation is conducted
according to the schemes shown in Figs. 6 and 7. Sample sizes are 1×104, 2×104, 4×104, 6×104, 8×104 and 1×105. The convergences of sensitivities for 10
tolerance factors are plotted in Fig. 8.

From the convergence of sensitivities for 10 tolerance factors, obtained in
Fig. 8, we can be found that the tolerance sensitivities convergence with
Halton-set based MC simulation, generally maintains a steadier process than
that with pseudo-random based one. By comparison, these statistics with
Halton-set based NTM simulation is more reliable than that with pseudo-random
in convergence precision. The final tolerance sensitivities, with a sample
size of 105 Halton-set based MC simulation is depicted in Fig. 9. The
analysis of variance is then carried out, to verify the tolerance
sensitivities as listed in Table 7.

Tolerance factor sensitivities have been yielded through range analysis in
Fig. 9. Variance analysis in Table 7 confirms that the orientation angular
errors, Δα and Δβ, for backplane of the L/R
mechanism are the most sensitive factors, and they should be tightly
controlled during the manufacturing process. Therefore, in the second stage
of DOE for tolerance assignment, the sensitive factors, Δα and
Δβ, are configured with top levels ±0.1∘ of
permitted tolerances; the insensitive ones such as Δxp, Δyp, Δzp, and Δai,x (or Δai,y), can be
assigned with a more relaxed tolerance ±0.5 mm; and the rest ones
including Δγ, Δai,z, Δθ1 (or Δθ2) and Δl with their permitted tolerance zones, are put
into the second stage of the uniform DOE for final tolerance assignment.

4.4.3 The Second stage of DOE

As discussed in Sect. 4.2, the more/less sensitive a factor is, the
narrower/broader an interval of adjacent levels should be in general.
Therefore, the interval for levels is expected to be inversely proportional
to their sensitivities. The uniform array of U20(44) is properly
enough by accommodating to 4 factors, 4 levels per factor and 20 combinations
of levels. The combinations of different levels among factors with
distributional regularity and uniformity, have an optimum coverage of the
tolerance space, and avoid enumeration that may possibly result in a
combination explosion. The levels re-divided for factors Δγ,
Δai,z, Δθ1 (or Δθ2) and Δl
are contained in Tables 8 and 9.

4.4.4 Tolerance assignment and evaluation

The newly updated levels of tolerance factors Δγ, Δai,z, Δθ1 (or Δθ2) and Δl, in
Tables 8 and 9, are put into the uniform array U20(44). With the
sensitive factors Δα and Δβ, and insensitive ones
Δxp, Δyp, Δzp, and Δai,x (or
Δai,y), the candidate tolerance assignments are formulated.
Consequently, the stack-ups as their responses, with a sample size of
105 Halton-set based MC simulation, are produced in Tables 10 and 11.

Generally, the complexity of product manufacturing can be lowered down, as
the tolerance is properly amplified. There may still be reachable compromise
between reliability and economy in tolerance assignments. We usually define
different functions to evaluate manufacturing consumption. Typical
cost-tolerance functions, such as linear, exponential, inverse square, and
power series one (Singh, 2009a, b), could precisely express the relations
between tolerance and consumption. When applied in a particularly compound
process, these types of cost-tolerance functions need to be modified with
experience.

Compared to tolerance determined by experience in tolerance relaxation, we
simply choose the linear accumulative tolerance as cost-tolerance function,
to make comparison of tolerance relaxation. Thus, the cost-tolerance
dimensional and angular functions, T∑d and
T∑a, can be defined as

(17)TΣd=Δxp+Δyp+Δzp+4(Δai,x+Δai,y+Δai,z)+4ΔlTΣa=Δα+Δβ+Δγ+4(Δθ1+Δθ2)

Relaxation of dimensional and angular tolerance, ATd and
ATa, can be evaluated as

(18)ATd=TΣ2d-TΣ1dTΣ1d×100%ATa=TΣ2a-TΣ1aTΣ1a×100%

where T∑1d and T∑1a denote the dimensional and angular
accumulated tolerance, respectively. They are originally determined by
experience in terms of Table 4, with T∑1d=3.8 mm and
T∑1a=1.65∘. T∑2d and
T∑2a of each assignment in rows in Tables 10 and 11 are
processed with Eq. (14), and their relaxation (ATd and
ATa) are finally tabulated.

Table 10Array U20(44) for responses with a sample size of 105
using Halton-set based MC simulation (part A).

4.5 Discussion

Tolerance assignments in Tables 10 and 11 in series with uniform DOE, provide
potentially feasible candidates with the application for the L/R mechanism
manufacturing. All the stack-ups of assignments as response, are upper than
99.73 % in terms of “6σ” principle. It indicates that all of them
satisfy manufacturing quality criteria. However, this does not mean all the
initial tolerance bounds restricted by capacity, can definitely generate
satisfying stack ups, since there may still be a few assignments unfeasible,
however, it will not disable tolerance synthesis.

The relaxation, ATd and ATa, of the tolerance
assignments with positive signs, indicate the tolerance of the assignment is
relaxed, compared to that of experience based tolerance assignment. Those
with negative sign, indicate assignments a contractive tolerance. However,
the majority of the mechanism's tolerances are significantly relaxed
considering: ATd is up to 144.7 % of the combination 9 in
Table 11, and ATa is up to 179.4 % of the combination 19 in
Table 11; even though there are seven combinations with negative relaxation
of ATa. Tolerance assignments with positive amplifications of
ATd and ATa as candidates can facilitate the
manufacturing process of the L/R mechanism.

In this research, the uniform DOE array used for tolerance assignments
generation (other than the other ones) presents a fact that the combinations
of the experiment have a uniform coverage of tolerance space. The uniformity
and representativeness can be proved mathematically. Both the Halton-set
based MC simulation for tolerance analysis, and uniform design used in
accuracy synthesis of the L/R mechanism, are applicable extensions of LDS in
NTM. The evenly scatted high-dimensional sampling points from NTM, have
promoted the accuracy analytical precision and efficiency. It demonstrates
NTM based MC approach prevails over traditional pseudo-random based one.
While the uniform DOE array, provides more representative tolerance
assignments, which can be in resistant to any tolerance applicative failure
induced by manufacturing system's capacity disturbance.

This research proposed DOE based tolerance synthesis approach for mechanism
tolerance synthesis, and applied for L/R mechanism with the application of
the SSMP. Conclusions are drawn as follows:

a.

A Halton-set based MC simulation is introduced and utilized in tolerance
synthesis process. Comparative results indicate that it could provide more
efficient and precise convergence in tolerance analysis.

b.

A DOE based tolerance synthesis is proposed. Sensitivities of all the
tolerance factors are revealed. Among them, the orientation angular errors
Δα and Δβ, of the backplane of the L/R mechanism,
are identified as the most significant factors, and validated through
variance analysis.

c.

Tolerance assignments, with a uniform coverage of tolerance space, through
uniform DOE are finally generated. The majority of assignments comply with
the manufacturing criteria of 99.73 %, and they have different tolerance
relaxation, compared to experience based tolerance assignment.

In the future research, tolerance synthesis will be extended to carry out
investigations on structural parameters, material selections and dynamic
response.

This research has been supported by the National Key R&D
Program of China (grant no. 2018YFB1304600), the CAS interdisciplinary
Innovation Team (grant no. JCTD-2018-11), the the Research Fund of China
Manned Space Engineering (grant no. 050102), the the Key Research Program of
the Chinese Academy of Sciences (grant no. Y4A3210301), and the the National
Science Foundation of China (grant nos. 51775541, 51175494, 51575412, and
61128008).